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ACT Math Review: Right-Triangle Trigonometry

Label the sides from the angle's point of view, pick the ratio, and solve.

sin θ = \frac{opposite}{hypotenuse}, cos θ = \frac{adjacent}{hypotenuse}, tan θ = \frac{opposite}{adjacent}

A right triangle has opposite side 3, adjacent side 4 and hypotenuse 5.

$sin θ = \frac{3}{5}$ , $cos θ = \frac{4}{5}$ , $tan θ = \frac{3}{4}$

A 13-foot ladder has its base 5 feet from a wall. How high does it reach?

$h^{2} + 5^{2} = 1 3^{2}$ , so $h = 12$ feet.

The angle with the ground has $cos θ = \frac{5}{13}$ .

You stand 60 feet from a tree, and the angle of elevation to the top is $3 5^{\circ}$ . How tall is the tree above eye level?

$tan 3 5^{\circ} = \frac{h}{60}$

$h = 60 tan 3 5^{\circ} \approx 42.0$ feet

Common mistake

Labeling sides before choosing the angle.

Opposite and adjacent depend on which angle you're using. The hypotenuse is always across from the right angle.

Try it yourself

If $sin θ = \frac{5}{13}$ in a right triangle, find $tan θ$ .
Show answer
$\frac{5}{12}$
A $6 0^{\circ}$ angle has an adjacent side of 8. Find the hypotenuse.
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16
Adjacent side 20, angle $4 0^{\circ}$ . Find the opposite side.
Show answer
$\approx 16.78$